Subgroups — C₂²×#19098804 #35155367

Order 20224.

3006 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14579140	C₂	1	2	yes	maximal
3	C₂×#19098804	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14579086	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₅₈	C₂²	1	2	yes	—
8	C₂²×Dic₃₁₆	—	2	8	no	—
9	C₂²×Dic₃₁₆	C₂²	1	12	yes	complemented
10	C₂×#14579140	—	2	12	no	—
11	C₂×#14579140	C₂²	1	24	yes	—
12	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂²×Dic₇₉⋊C₄	—	2	2	no	—
14	C₂²×C₂²⋊Dic₇₉	—	2	2	no	—
15	#19098804	C₂²	1	64	yes	direct product factor
16	C₂×#14579095	C₂²	1	12	yes	complemented
17	C₂×#14579086	C₂²	1	24	yes	complemented
18	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₉	—	2	8	no	—
20	C₂³×C₈×C₇₉	C₂²	1	12	yes	—
21	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₅₈	—	2	6	no	—
24	C₂²×Dic₁₅₈	—	2	8	no	—
25	C₂²×Dic₁₅₈	—	2	8	no	—
26	C₂²×Dic₁₅₈	C₂³	1	12	yes	—
27	C₂³×Dic₇₉	—	2	2	no	—
28	C₂³×Dic₇₉	—	2	2	no	—
29	C₂×Dic₃₁₆	—	2	48	no	—
30	C₂×Dic₃₁₆	C₂³	1	16	yes	complemented
31	#14579140	—	2	48	no	—
32	#14579140	C₂³	1	32	yes	—
33	C₂×C₄⋊Dic₇₉	—	2	6	no	—
34	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
35	C₂×Dic₇₉⋊C₄	—	2	24	no	—
36	C₂×C₂²⋊Dic₇₉	—	2	24	no	—
37	#14579095	C₂³	1	16	yes	complemented
38	#14579086	C₂³	1	32	yes	complemented
39	C₂⁵×C₇₉	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₉	—	2	4	no	—
41	C₂³×C₄×C₇₉	—	2	4	no	—
42	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₉	D₄	1	6	yes	—
44	C₂³×C₄×C₇₉	C₂³	1	6	yes	—
45	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	—	2	4	no	—
47	C₂²×C₈×C₇₉	—	2	48	no	—
48	C₂²×C₈×C₇₉	D₄	1	8	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	16	yes	—
50	C₂²×C₈×C₇₉	C₂³	1	6	yes	—
51	C₂²×C₈×C₇₉	C₂³	1	6	yes	—
52	C₂×Dic₁₅₈	—	2	24	no	—
53	C₂×Dic₁₅₈	—	2	12	no	—
54	C₂×Dic₁₅₈	—	2	24	no	—
55	C₂×Dic₁₅₈	—	4	24	no	—
56	C₂×Dic₁₅₈	C₂⁴	1	8	yes	—
57	C₂²×Dic₇₉	—	2	12	no	—
58	C₂²×Dic₇₉	—	2	12	no	—
59	C₂²×Dic₇₉	—	4	8	no	—
60	C₂²×Dic₇₉	—	4	8	no	—
61	Dic₃₁₆	—	2	32	no	—
62	C₄⋊Dic₇₉	—	2	12	no	—
63	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
64	Dic₇₉⋊C₄	—	2	32	no	—
65	C₂²⋊Dic₇₉	—	2	32	no	—
66	C₂⁴×C₇₉	—	2	4	no	—
67	C₂⁴×C₇₉	—	2	4	no	—
68	C₂⁴×C₇₉	Q₁₆	1	8	yes	complemented
69	C₂⁴×C₇₉	C₂×D₄	1	6	yes	—
70	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	—	2	4	no	—
72	C₂²×C₄×C₇₉	—	2	24	no	—
73	C₂²×C₄×C₇₉	—	2	12	no	—
74	C₂²×C₄×C₇₉	—	2	12	no	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₇₉	#41	1	4	yes	—
80	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
82	C₂×C₈×C₇₉	—	2	32	no	—
83	C₂×C₈×C₇₉	—	2	12	no	—
84	C₂×C₈×C₇₉	C₂×D₄	1	24	yes	—
85	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
86	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
87	C₂³×C₇₉	—	2	12	no	—
88	C₂³×C₇₉	—	2	12	no	—
89	C₂³×C₇₉	—	2	32	no	—
90	C₂³×C₇₉	#121	1	4	yes	—
91	C₂³×C₇₉	C₂×Q₁₆	1	4	yes	—
92	C₂³×C₇₉	C₂×Q₁₆	1	24	yes	complemented
93	C₂³×C₇₉	#135	1	4	yes	—
94	C₂³×C₇₉	C₂²×D₄	1	4	yes	—
95	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₇₉	—	2	12	no	—
97	C₂×C₄×C₇₉	—	2	24	no	—
98	C₂×C₄×C₇₉	—	2	4	no	—
99	C₂×C₄×C₇₉	—	2	4	no	—
100	C₂×C₄×C₇₉	#121	1	12	yes	—
101	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
103	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
104	C₂×C₄×C₇₉	C₂×#41	1	6	yes	—
105	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
106	C₂×Dic₇₉	—	2	8	no	—
107	C₂×Dic₇₉	—	2	8	no	—
108	C₂×Dic₇₉	—	4	24	no	—
109	C₂×Dic₇₉	—	4	24	no	—
110	Dic₁₅₈	—	2	8	no	—
111	Dic₁₅₈	—	2	8	no	—
112	Dic₁₅₈	—	4	24	no	—
113	C₈×C₇₉	—	2	4	no	—
114	C₈×C₇₉	C₂²×D₄	1	8	yes	—
115	C₂²×C₇₉	—	2	4	no	—
116	C₂²×C₇₉	—	2	48	no	—
117	C₂²×C₇₉	—	2	4	no	—
118	C₂²×C₇₉	#467	1	16	yes	complemented
119	C₂²×C₇₉	C₂×#121	1	6	yes	—
120	C₂²×C₇₉	C₂²×Q₁₆	1	6	yes	—
121	C₂²×C₇₉	C₂²×Q₁₆	1	8	yes	complemented
122	C₂²×C₇₉	C₂×#135	1	6	yes	—
123	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₇₉	—	2	4	no	—
125	C₄×C₇₉	C₂×#121	1	6	yes	—
126	C₄×C₇₉	C₂³×D₄	1	1	yes	characteristic
127	C₄×C₇₉	C₂²×#41	1	1	yes	characteristic
128	Dic₇₉	—	4	8	no	—
129	Dic₇₉	—	4	8	no	—
130	C₂²×#467	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	—	2	8	no	—
132	C₂×C₇₉	C₂×#467	1	12	yes	complemented
133	C₂×C₇₉	C₂²×#121	1	1	yes	characteristic
134	C₂×C₇₉	C₂³×Q₁₆	1	1	yes	characteristic
135	C₂×C₇₉	C₂²×#135	1	1	yes	characteristic
136	C₂³×Q₁₆	—	79	1	no	—
137	C₂⁴×C₈	D₇₉	1	1	yes	characteristic
138	C₂²×C₂²⋊Q₈	—	79	2	no	—
139	C₂×#467	—	79	24	no	—
140	C₂²×#107	—	79	1	no	—
141	C₂²×#103	—	79	2	no	—
142	C₇₉	C₂²×#467	1	1	yes	characteristic, complemented, Sylow
143	C₂³×Q₈	—	79	2	no	—
144	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
145	C₂²×Q₁₆	—	79	12	no	—
146	C₂²×Q₁₆	—	158	8	no	—
147	C₂³×C₈	—	2	8	no	—
148	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₇₉	1	12	yes	—
150	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
151	C₂×C₂²⋊Q₈	—	79	24	no	—
152	C₂×C₂²⋊Q₈	—	158	12	no	—
153	C₂²×C₄⋊C₄	—	79	2	no	—
154	C₂²×C₄⋊C₄	—	158	2	no	—
155	C₂²×C₂²⋊C₄	—	158	2	no	—
156	#467	—	79	64	no	—
157	C₂×#107	—	79	12	no	—
158	C₂×#103	—	79	24	no	—
159	C₂⁵	D₃₁₆	1	1	yes	characteristic
160	C₂²×Q₈	—	79	12	no	—
161	C₂²×Q₈	—	158	8	no	—
162	C₂²×Q₈	—	158	8	no	—
163	C₂²×Q₈	—	158	6	no	—
164	C₂³×C₄	—	2	4	no	—
165	C₂³×C₄	—	2	4	no	—
166	C₂³×C₄	—	158	2	no	—
167	C₂³×C₄	—	158	2	no	—
168	C₂³×C₄	D₃₁₆	1	6	yes	—
169	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₇₉	1	6	yes	—
171	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
172	C₂×Q₁₆	—	79	16	no	—
173	C₂×Q₁₆	—	158	48	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	48	no	—
176	C₂²×C₈	C₂²⋊D₇₉	1	8	yes	—
177	C₂²×C₈	C₂²×D₇₉	1	6	yes	—
178	C₂²×C₈	C₂²×D₇₉	1	16	yes	—
179	C₂²×C₈	C₂²×D₇₉	1	6	yes	—
180	C₂²⋊Q₈	—	79	32	no	—
181	C₂²⋊Q₈	—	158	48	no	—
182	C₂×C₄⋊C₄	—	79	12	no	—
183	C₂×C₄⋊C₄	—	158	6	no	—
184	C₂×C₄⋊C₄	—	158	24	no	—
185	C₂×C₂²⋊C₄	—	158	24	no	—
186	#107	—	79	16	no	—
187	#103	—	79	32	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	4	no	—
190	C₂⁴	Dic₃₁₆	1	8	yes	complemented
191	C₂⁴	C₂×D₃₁₆	1	6	yes	—
192	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
193	C₂×Q₈	—	316	24	no	—
194	C₂×Q₈	—	79	8	no	—
195	C₂×Q₈	—	158	24	no	—
196	C₂×Q₈	—	158	24	no	—
197	C₂×Q₈	—	158	12	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	12	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	316	8	no	—
203	C₂²×C₄	—	316	8	no	—
204	C₂²×C₄	—	158	12	no	—
205	C₂²×C₄	—	158	12	no	—
206	C₂²×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
207	C₂²×C₄	C₂×D₃₁₆	1	6	yes	—
208	C₂²×C₄	C₂×D₃₁₆	1	12	yes	—
209	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
210	C₂²×C₄	#11929624	1	4	yes	—
211	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
212	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
213	Q₁₆	—	158	32	no	—
214	C₂×C₈	—	2	12	no	—
215	C₂×C₈	—	2	32	no	—
216	C₂×C₈	C₂×C₂²⋊D₇₉	1	24	yes	—
217	C₂×C₈	C₂³×D₇₉	1	4	yes	—
218	C₂×C₈	C₂³×D₇₉	1	4	yes	—
219	C₄⋊C₄	—	79	8	no	—
220	C₄⋊C₄	—	158	32	no	—
221	C₄⋊C₄	—	158	12	no	—
222	C₂²⋊C₄	—	158	32	no	—
223	C₂³	—	2	12	no	—
224	C₂³	—	2	32	no	—
225	C₂³	—	2	12	no	—
226	C₂³	#14579135	1	4	yes	—
227	C₂³	C₂×Dic₃₁₆	1	4	yes	—
228	C₂³	C₂×Dic₃₁₆	1	24	yes	complemented
229	C₂³	#14579183	1	4	yes	—
230	C₂³	C₂²×D₃₁₆	1	4	yes	—
231	C₂³	C₂²×D₃₁₆	1	3	yes	—
232	Q₈	—	316	24	no	—
233	Q₈	—	158	8	no	—
234	Q₈	—	158	8	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	24	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	316	24	no	—
240	C₂×C₄	—	316	24	no	—
241	C₂×C₄	—	158	8	no	—
242	C₂×C₄	—	158	8	no	—
243	C₂×C₄	#14579135	1	12	yes	—
244	C₂×C₄	C₂²×C₂²⋊D₇₉	1	6	yes	—
245	C₂×C₄	C₂²×D₃₁₆	1	4	yes	—
246	C₂×C₄	C₂²×D₃₁₆	1	3	yes	—
247	C₂×C₄	C₂×#11929624	1	6	yes	—
248	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	C₂²×C₂²⋊D₇₉	1	8	yes	—
251	C₂²	—	2	48	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#19098804	1	16	yes	direct product factor
255	C₂²	C₂×#14579135	1	6	yes	—
256	C₂²	C₂²×Dic₃₁₆	1	6	yes	—
257	C₂²	C₂²×Dic₃₁₆	1	8	yes	complemented
258	C₂²	C₂×#14579183	1	6	yes	—
259	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	316	8	no	—
262	C₄	—	316	8	no	—
263	C₄	C₂×#14579135	1	6	yes	—
264	C₄	C₂³×C₂²⋊D₇₉	1	1	yes	characteristic
265	C₄	C₂²×#11929624	1	1	yes	characteristic
266	C₂	—	2	8	no	—
267	C₂	C₂×#19098804	1	12	yes	direct product factor
268	C₂	C₂²×#14579135	1	1	yes	characteristic
269	C₂	C₂³×Dic₃₁₆	1	1	yes	characteristic
270	C₂	C₂²×#14579183	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂⁴×C₈×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
prime radical	C₂⁵×C₇₉	D₄
hypercenter	C₂⁴×C₈	D₇₉
hypocenter	C₇₉	C₂²×#467
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×Dic₃₁₆	C₂	semidirect
3. C₂×#19098804	C₂	direct (N × H)
4. C₂²×#14579095	C₂	semidirect
5. C₂²×#14579086	C₂	semidirect
9. C₂²×Dic₃₁₆	C₂²	semidirect
15. #19098804	C₂²	direct (N × H)
16. C₂×#14579095	C₂²	semidirect
17. C₂×#14579086	C₂²	semidirect
30. C₂×Dic₃₁₆	C₂³	semidirect
37. #14579095	C₂³	semidirect
38. #14579086	C₂³	semidirect
68. C₂⁴×C₇₉	Q₁₆	semidirect
92. C₂³×C₇₉	C₂×Q₁₆	semidirect
118. C₂²×C₇₉	#467	semidirect
121. C₂²×C₇₉	C₂²×Q₁₆	semidirect
132. C₂×C₇₉	C₂×#467	semidirect
142. C₇₉	C₂²×#467	semidirect
190. C₂⁴	Dic₃₁₆	semidirect
228. C₂³	C₂×Dic₃₁₆	semidirect
254. C₂²	#19098804	direct (N × H)
257. C₂²	C₂²×Dic₃₁₆	semidirect
267. C₂	C₂×#19098804	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×Dic₃₁₆	2
C₂²×#14579140	2
C₂×#19098804	2
C₂²×#14579095	2
C₂²×#14579086	2
C₂⁴×C₈×C₇₉	2
C₂²×#467	79

Subgroups — C22×#19098804 #35155367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19098804 #35155367